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Polytope of Type {18,2,24}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,2,24}*1728
if this polytope has a name.
Group : SmallGroup(1728,15830)
Rank : 4
Schlafli Type : {18,2,24}
Number of vertices, edges, etc : 18, 18, 24, 24
Order of s0s1s2s3 : 72
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {9,2,24}*864, {18,2,12}*864
   3-fold quotients : {18,2,8}*576, {6,2,24}*576
   4-fold quotients : {9,2,12}*432, {18,2,6}*432
   6-fold quotients : {9,2,8}*288, {18,2,4}*288, {3,2,24}*288, {6,2,12}*288
   8-fold quotients : {9,2,6}*216, {18,2,3}*216
   9-fold quotients : {2,2,24}*192, {6,2,8}*192
   12-fold quotients : {9,2,4}*144, {18,2,2}*144, {3,2,12}*144, {6,2,6}*144
   16-fold quotients : {9,2,3}*108
   18-fold quotients : {3,2,8}*96, {2,2,12}*96, {6,2,4}*96
   24-fold quotients : {9,2,2}*72, {3,2,6}*72, {6,2,3}*72
   27-fold quotients : {2,2,8}*64
   36-fold quotients : {3,2,4}*48, {2,2,6}*48, {6,2,2}*48
   48-fold quotients : {3,2,3}*36
   54-fold quotients : {2,2,4}*32
   72-fold quotients : {2,2,3}*24, {3,2,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18);;
s1 := ( 1, 5)( 2, 3)( 4, 9)( 6, 7)( 8,13)(10,11)(12,17)(14,15)(16,18);;
s2 := (20,21)(22,23)(24,27)(25,29)(26,28)(30,33)(31,35)(32,34)(37,40)(38,39)
(41,42);;
s3 := (19,25)(20,22)(21,31)(23,26)(24,28)(27,37)(29,32)(30,34)(33,41)(35,38)
(36,39)(40,42);;
poly := Group([s0,s1,s2,s3]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(42)!( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18);
s1 := Sym(42)!( 1, 5)( 2, 3)( 4, 9)( 6, 7)( 8,13)(10,11)(12,17)(14,15)(16,18);
s2 := Sym(42)!(20,21)(22,23)(24,27)(25,29)(26,28)(30,33)(31,35)(32,34)(37,40)
(38,39)(41,42);
s3 := Sym(42)!(19,25)(20,22)(21,31)(23,26)(24,28)(27,37)(29,32)(30,34)(33,41)
(35,38)(36,39)(40,42);
poly := sub<Sym(42)|s0,s1,s2,s3>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 

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